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CHAPTER 13
asic Math Techniques
I. EXPONENTS AND SCIENTIFIC NOTATION
A. Exponents
B. Scientific Notation
II. LOGARITHMS
A. Common Logarithms
B. Antilogarithms
C. Natural Logarithms
D. An Application of Logarithms: pH
III. UNITS OF MEASUREMENT
IV. INTRODUCTION TO THE USE OF EQUATIONS
TO DESCRIBEA RELATIONSHIP
A. Equations
B. Units and Mathematical Operations
I.
EXPONENTS
AND SCIENTIFIC
NOTATION
EXAMPLE APPLICATION: EXPONENTS
SCIENTIFIC NOTATION
AND
a. The human genome consists of approximately 3 X 109
base pairs of DNA. If the sequence of the human
genome base pairs was written down, then it would
occupy 200 telephone books of 103 pages each. How
many base pairs would be on each page?
b. If it costs $0.50 to determine the location (sequence)
of a single base pair, how much would it cost to
determine the location of all 3 X 109 base pairs?
Answers on p. 222
A. Exponents
i.
THE MEANING OF EXPONENTS
We begin this chapter by discussing exponents and scientific notation. Exponents and scientific notation are
routinely used in the laboratory and so it is important
to be able to manipulate them easily and quickly.
An exponent is used to show that a number is to be
multiplied by itself a certain number of times. For example, 24 is read "two raised to the fourth power" and
means that 2 is multiplied by itself 4 times:
2
X
2
X
2
X
2
=
16
Similarly:
45
102 means: 10 X 10, or 100
means: 4 X 4 X 4 X 4 X 4 or 1024
The number that is multiplied is called the base and the
power to which the base is raised is the exponent. In the
expression 103, the base is 10 and the exponent is 3.
A negative exponent indicates that the reciprocal of
the number should be multiplied times itself. For
example:
1
1
1
1
3
10- = 10 X 10 X 10 = 1000 = 0.001
Rules that govern the manipulation of exponents in
calculations are summarized in Box 13. 1 on p. 219.
217
~~------------------------------------------------~~
218
FOUR
MATH
IN THE BIOTECHNOLOGY LABORATORY:AN OVERVIEW
Perform the operations
'" 106 '" one million
1,000,000
EXAMPLE PROBLEM
(6 places before d
100,000 ::: 105 '" one hundred thousand
4
1 0,000
indicated:
'" 1 0
'"
ten
=
= one
1 00 = 1 02 = one
1a = 10' = ten
1 ,000
10
1
0.1
(5 places before de --~
(4 places before decimal
thousand
1 03
{3 places before decimal pc--;
hundred
(2 places before decimal
10.1
=
(no places before decimal poin )
one tenth
(1 place right of the decimal pol
= one hundredth
0.001 = 10-3 = one thousandth
0.01
0.0001
ANSWER
ii.
10
1849 + 2197
EXPONENTS WHERE THE BASE
10-2
=
10-4
=
(2 places right of the decimal point]
(3 places right of the decimal poi
one ten thousandth
(4 places right of the decimal poin
= 10-5 = one hundred thousandth
0.000001 = 10-6 = one millionth
(5 places right of the decimal point)
0.00001
The denominator involves addition of numbers with
exponents. Convert the numbers with exponents to
standard notation. Then perform the calculations.
10
=
=~
4046
= 0.0025
Is 10
In preparation for discussing scientific notation, consider the particular case of exponents where the base is
10. Observe the following rules as illustrated in Figure
13.1:
1. For numbers greater than 1:
• the exponent represents the number of places
after the number (and before the decimal point)
• the exponent is positive
• the larger the positive exponent, the larger the
number
2. For numbers less than 1:
• the exponent represents the number of places to
the right of the decimal point including the first
nonzero digit
• the exponent is negative
• the larger the negative exponent, the smaller
the number
People commonly use the phrase orders of magnitude where one order of magnitude is 101. Using this
terminology, 102 is said to be two orders of magnitude
less than 104. Similarly, 108 is three orders of magnitude
greater than 105.
B. Scientific Notation
i. EXPRESSING NUMBERS IN SCIENTIFIC NOTATION
cientific notation is a tool that uses exponents to simplify handling numbers that are very big or very small.
Con ider the number:
0.0000000000000000000000602
pour
(1 place before decimal poin")
= 10° = one
=
---
thousand
(6 places right of the decimal point)
Figure 13.1. Using Exponents Where the Base Is 10.
(When the decimal point is not written it is assumed to be to
the right of the final digit in the number.)
In scientific notation this lengthy number is compactly
expressed as:
6.02 X 10-23
A value in scientific notation is customarily written
as a number between 1 and 10 multiplied by 10 raised
to a power. For example:
100 (Standard
Notation) =
1
2
10
X
=
102 = 1 X 102 (Scientific
1 X 10 X 10 = 100
300 (Standard Notation) = 3 X 102 (Scientific
3 X 102 = 3 X 10 X 10 = 300
Notation)
Notation)
The number of bacterial cells in 1 liter of culture might be
100,000,000,000 (Standard Notation) =
1 X 1011(Scientific Notation)
A number in scientific notation has two parts. The
first part is sometimes called the coefficient. The second
part is 10 raised to some power, the exponential term.
For example:
Second Part
First Part
(Coefficient)
(Exponential Term)
1000 =
1
X
103
X
102
235 =
2.35
As shown in Figure 13.1, a negative exponent is used
for a number less than 1. Three examples:
1
X
10-5
=
.l. X
~
10
10
X ~
X
.l. X
~
10
10
10
1
100, 000 = 0.00001
0.000135 = 1.35 X 10-4
A bacterial cell wall is about 0.00000001 m
= 1 X 10-8 m thick
A procedure to convert a number from standard notation to scientific notation is shown in Box 13.2 on p. 220.
EXAMPLE PROBLEM
Convert the number 0.000348
to scientific notation.
ANSWER
Step I. This number is less than I. Move the decimal
point to the right: 0.000348 --* 3.48
u..u1
CHAPTER
Box 13.1.
CALCULATIONS
INVOLVING
I3
BASIC MA
rEG-!
zs
ExPONENTS
1. To multiply two numbers with exponents where the numbers have the same base, add the exponents:
53 X 56 = 59
10-3 X 104 = 101
Two examples:
To convince yourself that this rule makes sense; consider the following example:
23 X 22 = 25
= (2
X
2
X
2)
(2
X
X
2)
=
2 multiplied 5 times = 2 = 32
5
2. To divide two numbers with exponents where the numbers have the same base, subtract the exponents:
53/56 = 53-6 = 5-3
2-3/2-4 = 2(-3)-(-4) = 21 = 2
Two examples:
You can convince yourself that this rule makes sense by rewriting an example this way:
5356
/
=
% X% X%
%x%x%x5x5x5
5
X
1
5
=
X
5
_1_
125
=
5-3
3. To raise an exponential number to a higher power, multiply the two, exponents.
(23)2 = 26
(103)-4 = 10-12
Two examples:
To convince yourself that this rule makes sense, examine this example:
(23)2 = 23 X 23 = (2 X 2 X 2) X (2 X 2 X 2) = 26
4. To multiply or divide numbers with exponents that have different bases, convert the numbers with exponents
to their corresponding values without exponents. Then, multiply or divide.
Multiply: 32
Two examples:
X
24
3 = 9 and 24 = 16,
2
so 9 X 16
4 -3
=
=
144
Divide: 4-3/23
1
1
1
1
- X - X - = - = 0.015625
4
4
4
64
and 23
so
=
8
0.015625 _ 0 00 5
8
-.
19
-. To add or subtract numbers with exponents (whether their bases are the same or not), convert the numbers
with exponents to their corresponding values without exponents.
For example:
43 + 23 = 64 + 8 = 72
. By definition, any number raised to the 0 power is 1.
For example:
85°
=1
~OU
Box 13.2.
A
MATH IN THE BIOTECHNOLOGY LABORATORy:AN OVERVIEW
PROCEDURE
TO CONVERT
A NUMBER
FROM STANDARD
NOTATION
TO SCIENTIFIC
NOTATION
tep 1. a. If the number in standard notation is greater than 10, then move the decimal point to the left so ilia
there is one nonzero digit to the left of the decimal point. This gives the first part of the notation.
b. If the number in standard notation is less than 1, then move the decimal point to the right so that there
is one nonzero digit to the left of the decimal point. This gives the first part of the notation.
c. If the number in standard notation is between 1 and 10, then scientific notation is seldom used.
Step 2. Count how many places the decimal was moved in step 1.
Step 3. a. If the decimal was moved to the left, then the number of places it was moved gives the exponent in the
second part of the notation.
b. If the decimal point was moved to the right, then place a - sign in front of the value. This is the exponent for the second part of the notation.
EXAMPLE
Express the number 5467 in scientific notation.
Step 1. This number is greater than 10. Therefore, move the decimal point to the left so that there is only one
nonzero digit to the left of the decimal point:
5467. move decimal point 3 places left ---+ 5.467
=
tJ...u
Step 2. The decimal point was moved three places to the left.
Step 3. The exponent for the second part of the notation is therefore 3. This means the number in scientific notation
is: 5.467 x 103
Step 2. The decimal point was moved four places to the
right.
I. Convert the expression on the left to standard notation; it equals 55.
Step 3. The exponent is -4, so the answer in scientific
notation is: 3.48 X 10-4
2. The number on the right side of the expression, that
is, 5500, is larger than the number on the left, that is,
55. The exponent that fills in the blank, therefore, will
need to be a negative number (to make 5500 smaller).
So far, we have shown the customary manner of writing numbers in scientific notation, that is, with the coefficient written as a number between 1 and 10. It it is not
necessary, however, always to write numbers in scientific notation in this way. For example, the value 205
may be expressed as:
205.
205.
205.
205.
205.
=
=
=
=
=
X 103
0.205
2.05 X 102
20.5 X 101
2050 X 10-1
20500 X 10-2
Similarly:
1.00 X 104 = 10.0 X 103 = 100. X 102
3.45 X 1023 = 0.0345 X 1025 = 345 X 1021
There are situations where it is useful to manipulate
coefficients and exponents without changing the value
of the numbers. This is the case in addition and subtraction of numbers expressed in scientific notation.
EXAMPLE PROBLEM
Fill in the blank so that the numbers on both sides of
the = sign are equal. (For example: 2.58 X 10-2 =
25.8 X 10-3)
0.0055
X 104
=
5500
X
ANSWER
One way to think about this problem:
10-
3. 5500 times 10-2 equals 55.The answer is therefore - 2.
ii.
CALCULATIONS
WITH
SCIENTIFIC
Box 13.3 summarizes methods
tions with scientific notation.
NOTATION
of performing
calcula-
EXAMPLE PROBLEM
(3.45
X
1023)
+ (4.56
X
1025)
=
?
ANSWER
The two numbers in this example would require many
zeros if written in standard notation. The strategy of
expressing both values in scientific notation with the
same exponent, therefore, is preferred over converting
both numbers to standard notation.
Step I. Decide on a common exponent. Suppose we
choose 25.
Step 2. Express both numbers in a form with the same
exponent.
3.45 X 1023 = 0.0345 X 1025
Step 3. Perform the addition.
0.0345 X 1025
+
4.56
X
1025
4.5945
X
1025
(Note: If the coefficient is rounded according to significam figure
ventions, then the answer is 4.59 x 1025)
CHAPTER
Box 13.3.
CALCULATIONS
INVOLVING
NUMBERS
IN SCIENTIFIC
13
21
BASIC MATH TECH IQIF
NOTATION
1. To multiply numbers in scientific notation use two steps:
Step 1. Multiply the coefficients together.
Step 2. Add the exponents to which 10 is raised.
For example:
(2.34
X
102) (3.50
103)
X
(multiply the coefficients)
(add the exponents)
102+3
(2.34 X 3.50)
X
=
= 8.19
X
105
2. To divide numbers in scientific notation, use two steps:
Step 1. Divide the coefficients.
Step 2. Subtract the exponents to which 10 is raised.
For example:
(subtract the exponents)
(divide the coefficients)
(4.5/2.1)
105-3
X
= 2.1 X 102
3. To add or subtract numbers in scientific notation:
a. If the numbers being added or subtracted all have 10 raised to the same exponent, then the numbers can be
simply added or subtracted as shown in these examples:
(3.0 X 104) + (2.5 X 104) = ?
(7.56 X 1021) - (6.53 X 1021) = ?
+
3.0
2.5
5.5
X
X
X
104
104
104
-
7.56
6.53
1.03
X
X
X
1021
1021
1021
b. If the numbers being added or subtracted do not all have 10 raised to the same exponent, then there are two
strategies for adding and subtracting numbers.
.
STRATEGY 1 Convert the numbers to standard notation and then do the addition or subtraction:
(2.05
For example:
X
102)
-
(9.05
X
10-1)
=
?
Convert both numbers to standard notation:
2.05 X 102
9.05 X 10-1
Perform the calculation:
= 205
= 0.905
205
0.905
204.095
(Note: If this value is rounded according to significant figure conventions, then the answer is 204.)
STRATEGY 2 Rewrite the values so they all have 10 raised to the same power:
(2.05
For example:
X
102)
-
(9.05
X
10-1)
=
?
To convert both numbers to a form such that they both have 10 raised to the same power:
Step 1. Decide what the common exponent will be. It should be either 2 or -1.
Suppose we choose 2.
Step 2. Convert 9.05
X
10-1 to a number in scientific notation with the exponent of 2:
9.05
Step 3. Perform the subtraction:
X
10-1
= 0.00905
2.05
- 0.00905
2.04095
X
X
X
X
102.
102
102
102
( ote: If the coefficient is rounded according to significant figure conventions, then the answer is 2.04
X 1()2.)
IN THE BIOTECHNOLOGY
MATH
LABORATORY: AN OVERVIEW
A cal ulator will hold a limited number
of places and
not a ept very large or very small numbers if you
;::-y 0 -ey in all the zeros. A scientific calculator, how~ - T. works
easily with large and small numbers
expre sed in scientific notation. On my calculator, to
key in the number 1 X 103, I push the following keys:
1
exp
3
3. Underline the larger number in each pair.
both if their values are equal.
a. 5 X 10-3 em, 500 X 10-1 em
10-3 IJL,3000
c. 3.200 X 10-6 m, 3200
b. 300
X
X
nderlin
10-2 IJL
10-4 m
3
em, 1 X 10- em
3
10- L, 0.0008 X 10-4 L
d. 0.001
X 101
e. 0.008
X
X
4. Convert the following numbers
to scientific notation.
_.ote that I do not key in the number 10; the base in sci-
a. 54.0
b. 4567
c. 0.345000
d. 10,000,000
entific notation. "exp" tells my calculator that the 10 is
present. To key in the number 3 X 10-4 on my calculator, I press:
e. 0.009078
f. 540
g. 0.003040
h. 200,567,987
3
exp
4
5. Convert the following numbers to standard notation.
a. 12.3 X 103
b. 4.56 X 104
c. 4.456 X 10-5
d. 2.300 X 10-3
e. 0.56 X 106
f. 0.45 X 10-2
6. Perform
+/The +/- key tells the calculator I want a negative exponent. "EE" is sometimes used to indicate that 10 is being
raised to a certain power. Consult your instruction manual to see how to key in a number in scientific notation.
EXAMPLE APPLICATION
ANSWER (from p.217)
a. The total number of pages needed to record the
genome would be
200
X
103
=
2
X
105
=
200,000
Then, the number of base pairs on each page would
be:
9
3 X 10
---..,.
=
5
2 X 10
1.5
X
104
=
a.
(4.725
(1.93
b. 3 X 109 X $0.50 = 1.5 X 109 = 1.5 billion dollars!
(There has been much effort in reducing the cost of
sequencing DNA.)
MANIPULATION
PRACTICE PROBLEMS:
EXPONENTS AND SCIENTIFIC NOTATION
the operations indicated:
a. 22 X 33
b. (143)(36)
C. 55 - 23
d. 57/84
e. (6-2)(32)
f. (-0.4)3
10-2)
X
b.
(8.8 X 108)(6.0 X 10-6)
c. (4.5 X 103) + (2.7 X 10-2)
e. (5.4
X
104)
1024)
f. (5.7
X
10-3) - (3.4
d. (35.6
X
-
(54.6
+ (3.4
106)
1026)
X
X
X
10-6)
7. Fill in the blanks so that the numbers on both sides
of the = sign are equal. For example: 2.58 X 10-2 =
25.8 X 10-3
a. 0.0050 X 10-4 = 0.050 X 10- = 0.50 X 105.0
c. 5.45
X
10-
10-3 =
X 101
10-3 = 54.5
X
X
d. 100.00
e. 6.78
X
f. 54.6
X
g. 45.6
X
h. 4.5
1. Give the whole number or fraction that corresponds
to these exponentiai expressions.
a. 22
b. 33
C. 2-2
d. 3-:-3
e. 102
f. 104
g. 10-2
h. 10-4
i. 5°
103)(4.22
X
b. 15.0
'
108)(0.0200)
X
(3700)(0.770)
=
15 000
the following calculations.
X
X
X
10-2
X
10-1
10-
101 = 1.0000 X 102
10 = 0.678 X 10102 =
X 106
X
108
10-3
= __
X
106
=
X
10-5
10-3
101.
i. 356.98
X
j. 0.0098
X 10-2 = 0.98 X 10-
=
X
II. LOGARITHMS
2. Perform
g. a2 X a3
j. (c '
3
r
5
i. (34)2
h. c3/C-6
k. 432
13
+
133
2
I. 10 /10
3
EXAMPLE APPLICATION:
+ 9.62
LOGARITHMS
The concentration of hydrochloric acid, Hel, secreted by
the stomach after a meal is about 1.2 X 10-3 M.What is
the pH of stomach acid?
Answer on p. 225